Statistical Downscaling Based on Spartan Spatial ...

T.U.C

Statistical Downscaling Based on Spartan Spatial Random Fields Dionissios T. Hristopulos Geostatistics Research Unit, T.U.C., Chania 73100 GREECE

Technical University of Crete

Abstract Stochastic methods of space-time interpolation and conditional simulation are used in statistical downscaling approaches to increase the resolution of measured fields. A popular interpolation method is kriging, also known as optimal interpolation, in data assimilation. Kriging is a stochastic, linear interpolator which incorporates time/space variability by means of the variogram function. However, estimation of the variogram involves various assumptions and simplifications, and the O(N 3) numerical complexity of kriging slows down performance for very large data sets. We present a different approach based on Spartan Spatial Random Fields (SSRFs). SSRFs were motivated from classical field theories of statistical physics [1]. The SSRFs model spatial dependence based on effective interactions, which can be formulated based on general statistical principles or physical constraints. The SSRF framework leads to a new family of covariance functions [2] and to new perspectives in covariance parameter estimation and interpolation [3]. A significant advantage of SSRFs is reduced numerical complexity, which can improve the speed of spatial interpolation and conditional simulation. On grids composed of rectangular cells, SSRFs lead to explicit expressions for the precision matrix. Therefore SSRFs constitute useful models of covariance functions for data assimilation methods. We use real GDR radioactivity measurements to demonstrate SSRF properties and downscaled fields.

Introduction

SSRF Method for Downscaling by Conditional Simulation

Variational methods are widely used in weather forecasting, hydrology and oceanography to combine observations with model background states into more accurate estimates of the actual state [4]. In variational data assimilation, a cost function measures the fit of the model to the assimilated observations and background state. For observations that are uncorrelated with the background, the cost function involves an observation term and a background term. The precision matrices (inverse of the covariance matrices) specify the weights used in the cost function. For example, in 3D-Var J(z) = (z − zb)T B−1(z − zb) + (w − H[z])T R−1(w − H[z]),

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The inverse of the error covariance matrix of the observations and that of the background state define the respective weighting matrix for each term.

Conclusions

I Sampling

set (S): 200 measurements of background ground dose rate (GDR) from Germany (used in the Spatial Interpolation Comparison 2004 exercise). Units: nanoSievert per hour (nSv/h). i h
λ I Data are normalized using a modified Box-Cox transform: Z → Z − Zmin + α2 − 1 /λ. SSRF model parameters are determined using the modified method of moments: η0 = 6.7409 104(nSv/h)−2, η1 = 2.0299, ξ = 41.2581 km, kc = 0.1485 (km)−1. The map grid (G) is a rectangle with 300 × 300 nodes.  1/2 ˜ z(k) ˜ I Unconstrained simulation is based on the spectral method, i.e., Z(k) uk, where u ∼ N(0, 1). = C I The

I Conditional

simulation is performed by combining unconstrained simulations with SSRF interpolation: ˆ → G) − Z ˆ nc(S → G). Zc(G) = Znc(G) + Z(S I Computational time (per realization) is ≈20 seconds. Matlabr 7.9.0 (R2009b) programming environment running on a desktop with an Intel(R) Xeon(R) CPU, 2.83GHz with 16GB of RAM.

Results of Gamma Dose Rate Downscaling Simulations

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SSRF Covariance Spectral Density

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(a) Unconstrained SSRF simulations.

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1 ξ2 ξ4 Jz(si, sj; θ) = J0(si, sj) + η1 2 J1(si, sj) + 4 J2(si, sj) , η0 ξ α α

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J0(si, sj) = δi,j: identity matrix; J1(si, sj): gradient matrix;

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Appendix: SSRF Precision Matrix (1D) [5]

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> 0: scale coefficient I η1 > −2: stiffness coefficient I ξ: characteristic length I kc: cutoff wavenumber (spectral band limit)

ˇ M. Zukoviˇ c and D. T. Hristopulos. Environmental time series interpolation based on spartan random processes. Atmospheric Environment, 42(33):7669–7678, 2008.

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A. R. Robinson and P.F.J. Lermusiaux. Data assimilation in models. In Encyclopedia of Ocean Sciences, pages 623–634. Academic Press Ltd., London, UK, 2001.

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Geostatistics Research Unit

D. T. Hristopulos. Spartan Gibbs random field models for geostatistical applications. SIAM Journal of Scientific Computing, 24(6):2125–2162, 2003.

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d η ξ h(kc − |k| 0 ˜ Cz(k; θ) = 1 + η1 (k ξ)2 + (k ξ)4 Square lattice of step a: d η ξ 0 ˜ z(k; θ) = C Pd 4 ξ 2 2 aki Pd 16 ξ 4 4 aki 1 + η1 i=1 a2 sin ( 2 ) + i=1 a4 sin ( 2 )

References

D. T. Hristopulos and S. N. Elogne. Computationally efficient spatial interpolators based on Spartan spatial random fields. IEEE Transactions on Signal Processing, 57(9):3475–3487, 2009. 600

 2 2 2 2 S0 = [Z(s)] , S1 = [∇Z(s)] , S2 = ∇ Z(s) .

permit computationally fast conditional simulations for downscaling meteorological fields I At this point, deviations from Gaussianity are handled by means of the Box-Cox transform I Downscaling simulations are presented for an irregularly sampled, slightly non-Gaussian GDR data set

D. T. Hristopulos and S. Elogne. Analytic properties and covariance functions of a new class of generalized Gibbs random fields. IEEE Transactions on Information Theory, 53(12):4667–4679, 2007.

PDF of MultiGaussian SSRF - Continuum

ffgc ∝ exp {−Hfgc[Z(s); θ]} , θ = (η0, η1, ξ, kc). Z  1 2 4 ds S0[Z(s)] + η1 ξ S1[Z(s)] + ξ S2[Z(s)] , Hfgc = d 2η0ξ

I SSRFs

(b) Conditional SSRF simulations

(c) Mean values and standard deviation.

Figure: Results of SSRF downscaling of the GDR data set on a 300 × 300 grid. (a) Ten unconstrained GDR realizations. The statistical properties of the data are captured but not the local information. (b): Ten conditional simulations constrained on 200 data. Both statistical information and local values are honored. (c) Mean and standard deviation based on 1000 simulations. The sampling sites are marked by black dots on the standard deviation map and have zero variance.

J2(si, sj) curvature matrix 

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1 −1 0 · · · 0  −1 2 −1 · · · 0    ... ... ... , J1(si, sj) =     0 · · · −1 2 −1  0 · · · 0 −1 1   1 −2 1 0 · · · 0  −2 5 −4 1 · · · 0     1 −4 6 −4 1 0  . J2(si, sj) =  ... ... ... ...      0 · · · 1 −4 5 −2  0 · · · 0 1 −2 1

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Technical University of Crete